It appears that in the films almost every single character uses a wand to cast spells, but then we have seen a man in the leaky cauldron stirring a cup of coffee without an obvious use of a wand or physical interaction. This suggests to me that it is possible for wizards to cast spells without wands, it's just a lot harder. So how does having a wand help?
The man that appears in the leaky cauldron.
Like you mentioned. It is a lot easier with wands. I doubt any major spell can be performed without the use of wands.
Also, it seems only in the movies that magic can occur without a wand. There is no evidence in the books to suggest that fact. In fact, all the evidence in the books suggests that witches and wizards are powerless without the wands. That's why even powerful wizards like Dumbledore and Voldemort use their wands all the time. Considering Voldemort discovered how to fly, you would think he would find a way to perform all kinds of magic without needing to hold on to a piece of wood, if it was possible.
While I am no expert on the movies (IMO, they are an insult to the books), it could just be that the wizard in question has cast a spell to make the spoon follow the directions of his finger. Though, if this was the case, I don't understand why he would not just cast a spell to make the spoon stir on its own without the need of a finger to guide it.
